منابع مشابه
Smith Normal Form and acyclic matrices
An approach, based on the Smith Normal Form, is introduced to study the spectra of symmetric matrices with a given graph. The approach serves well to explain how the path cover number (resp. diameter of a tree T ) is related to the maximal multiplicity MaxMult(T ) occurring for an eigenvalue of a symmetric matrix whose graph is T (resp. the minimal number q(T ) of distinct eigenvalues over the ...
متن کاملOn Positive Matrices Which Have a Positive Smith Normal Form
It is known that any symmetric matrix M with entries in R[x] and which is positive semi-definite for any substitution of x ∈ R, has a Smith normal form whose diagonal coefficients are constant sign polynomials in R[x]. We generalize this result by considering a symmetric matrix M with entries in a formally real principal domain A, we assume that M is positive semi-definite for any ordering on A...
متن کاملSmith normal form and Laplacians
Let M denote the Laplacian matrix of a graph G. Associated with G is a finite group Φ(G), obtained from the Smith normal form of M , and whose order is the number of spanning trees of G. We provide some general results on the relationship between the eigenvalues of M and the structure of Φ(G), and address the question of how often the group Φ(G) is cyclic.
متن کاملSmith normal form in combinatorics
This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith normal form of random integer matrices. We then give some examples of Smith normal form and diagonal form arising from (1) symmetric functions, (2) a result of Ca...
متن کاملOrthogonal Polynomials and Smith Normal Form
Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt–Stanley results for the Smith normal form of a certain multivariate matrix that refines one studied by Berlekamp, Carlitz, Roselle, and Scoville. The second argument, which uses orthogonal polynomials, generalizes to a number...
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ژورنال
عنوان ژورنال: Journal of Algebraic Combinatorics
سال: 2008
ISSN: 0925-9899,1572-9192
DOI: 10.1007/s10801-008-0121-8